Viviani, Vincenzo, De maximis et minimis, geometrica divinatio : in qvintvm Conicorvm Apollonii Pergaei

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        <div xml:id="echoid-div222" type="section" level="1" n="101">
          <head xml:id="echoid-head106" xml:space="preserve">LEMMA IV. PROP. XXXIX.</head>
          <p>
            <s xml:id="echoid-s2356" xml:space="preserve">Si duæ menſales ABCD, EFGH fuerint ſuper eadem linea AH
              <lb/>
            ad eaſdem partes deſcriptæ, ita vt ipſarum baſes AB, DC; </s>
            <s xml:id="echoid-s2357" xml:space="preserve">EF,
              <lb/>
            HG ſint omnes inter ſe parallelæ, ſintque proportionales lateribus
              <lb/>
            in directum poſitis; </s>
            <s xml:id="echoid-s2358" xml:space="preserve">nempe ſit vt AB ad EF, ita AD ad EH, & </s>
            <s xml:id="echoid-s2359" xml:space="preserve">DC
              <lb/>
            ad HG. </s>
            <s xml:id="echoid-s2360" xml:space="preserve">Dico, & </s>
            <s xml:id="echoid-s2361" xml:space="preserve">reliqua latera BC, FG eſſe inter ſe parallela.</s>
            <s xml:id="echoid-s2362" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2363" xml:space="preserve">DVctis enim diagonalibus BD, FH, productaque BC in I. </s>
            <s xml:id="echoid-s2364" xml:space="preserve">Cum ſit BA
              <lb/>
            ad EF, vt AD ad EH, erit permutando BA ad AD, vt FE ad EH, eſtq;
              <lb/>
            </s>
            <s xml:id="echoid-s2365" xml:space="preserve">angulus BAD ęqualis angulo FEH, ob parallelas BA, FE, quare triangulum
              <lb/>
            BAD ſimile eſt triangulo FEH; </s>
            <s xml:id="echoid-s2366" xml:space="preserve">ideoq; </s>
            <s xml:id="echoid-s2367" xml:space="preserve">angulus BDA æqualis angulo FHE,
              <lb/>
            & </s>
            <s xml:id="echoid-s2368" xml:space="preserve">totus CD A, æquatur
              <lb/>
              <figure xlink:label="fig-0092-01" xlink:href="fig-0092-01a" number="62">
                <image file="0092-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0092-01"/>
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            toto GHE, ob parallelas
              <lb/>
            CD, GH, vnde reliquus
              <lb/>
            CDB, æquatur reliquo
              <lb/>
            GHF. </s>
            <s xml:id="echoid-s2369" xml:space="preserve">Item cum ſit CD
              <lb/>
            ad GH, vt DA ad HE,
              <lb/>
            erit permutando CD ad
              <lb/>
            DA, vt GH ad HE, eſtq;
              <lb/>
            </s>
            <s xml:id="echoid-s2370" xml:space="preserve">DA ad DB, vt HE ad HF,
              <lb/>
            ob triangulorum ADB,
              <lb/>
            EHF ſimilitudinem, qua-
              <lb/>
            re ex æquali CD ad DB,
              <lb/>
            erit vt GH ad HF, ſuntq; </s>
            <s xml:id="echoid-s2371" xml:space="preserve">anguli ad D, & </s>
            <s xml:id="echoid-s2372" xml:space="preserve">H æquales, ergo, & </s>
            <s xml:id="echoid-s2373" xml:space="preserve">angulus D CB,
              <lb/>
            ſiue HIB, æqualis angulo HGF, ideoque rectæ BC, FG inter ſe æquidiſtant. </s>
            <s xml:id="echoid-s2374" xml:space="preserve">
              <lb/>
            Quod erat, &</s>
            <s xml:id="echoid-s2375" xml:space="preserve">c.</s>
            <s xml:id="echoid-s2376" xml:space="preserve"/>
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        <div xml:id="echoid-div224" type="section" level="1" n="102">
          <head xml:id="echoid-head107" xml:space="preserve">THEOR. XX. PROP. XXXX.</head>
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            <s xml:id="echoid-s2377" xml:space="preserve">Sià terminis æqualium ſegmentorum ex diametris ſimilium Hy-
              <lb/>
            perbolarum abſciſſorum rectæ ordinatim applicentur, vſque
              <lb/>
            ad ſectionum aſymptoto
              <unsure/>
            , erit ſegmentum applicatæ in Hyperbo-
              <lb/>
            la maiorum laterum, inter ſectionem, & </s>
            <s xml:id="echoid-s2378" xml:space="preserve">aſymptoton interceptum,
              <lb/>
            maius ſegmento applicatæ, quod in Hyperbola minorum laterum
              <lb/>
            inter ſectionem eiuſque aſymptoton intercipitur.</s>
            <s xml:id="echoid-s2379" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2380" xml:space="preserve">ESto Hyperbole maiorum laterum ABC, cuius tranſuerſum DB rectum
              <lb/>
            BE, aſymptotos FG, & </s>
            <s xml:id="echoid-s2381" xml:space="preserve">Hyperbole minorum ſit HIL, cuius tranſuer-
              <lb/>
            ſum MG, rectum GN, aſymptotos OP, & </s>
            <s xml:id="echoid-s2382" xml:space="preserve">ipſarum ſectionum diametris, dem-
              <lb/>
            pta ſint ęqualia diametri ſegmenta BQ, IR, è quorũ terminis Q, R applicate
              <lb/>
            ſint (ad partes æqualium inclinationum) rectæ QAG, RHP vique ad earum
              <lb/>
            aſymptotos: </s>
            <s xml:id="echoid-s2383" xml:space="preserve">Dico ſegmentum GA in ſectione maiorum laterum, maius eſſe
              <lb/>
            ſegmento PH in ſectione minorum.</s>
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